Properties

Label 2-504-21.5-c3-0-11
Degree $2$
Conductor $504$
Sign $0.874 + 0.485i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.34 + 5.79i)5-s + (−12.7 − 13.4i)7-s + (28.2 − 16.3i)11-s + 67.9i·13-s + (−15.3 − 26.5i)17-s + (−21.8 − 12.6i)19-s + (−68.6 − 39.6i)23-s + (40.0 + 69.4i)25-s − 109. i·29-s + (238. − 137. i)31-s + (120. − 28.8i)35-s + (160. − 277. i)37-s − 184.·41-s + 364.·43-s + (25.7 − 44.6i)47-s + ⋯
L(s)  = 1  + (−0.299 + 0.518i)5-s + (−0.687 − 0.725i)7-s + (0.775 − 0.447i)11-s + 1.44i·13-s + (−0.218 − 0.378i)17-s + (−0.264 − 0.152i)19-s + (−0.622 − 0.359i)23-s + (0.320 + 0.555i)25-s − 0.702i·29-s + (1.38 − 0.797i)31-s + (0.582 − 0.139i)35-s + (0.711 − 1.23i)37-s − 0.704·41-s + 1.29·43-s + (0.0799 − 0.138i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.564508461\)
\(L(\frac12)\) \(\approx\) \(1.564508461\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (12.7 + 13.4i)T \)
good5 \( 1 + (3.34 - 5.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.2 + 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 67.9iT - 2.19e3T^{2} \)
17 \( 1 + (15.3 + 26.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (21.8 + 12.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (68.6 + 39.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 109. iT - 2.43e4T^{2} \)
31 \( 1 + (-238. + 137. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-160. + 277. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 184.T + 6.89e4T^{2} \)
43 \( 1 - 364.T + 7.95e4T^{2} \)
47 \( 1 + (-25.7 + 44.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-532. + 307. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-207. - 359. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-411. - 237. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-142. - 246. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 965. iT - 3.57e5T^{2} \)
73 \( 1 + (-225. + 130. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-219. + 379. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 76.4T + 5.71e5T^{2} \)
89 \( 1 + (-356. + 617. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 410. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44558110013849284949411139337, −9.546193454981735346504452296717, −8.793375057189476438995977955704, −7.52094607241001194267519706714, −6.75429112234377824517644290135, −6.08548741948031402476878554534, −4.37239785047528977586597488580, −3.74189031449812352863697185130, −2.36608479775472213905830294554, −0.64700257620242349491189483054, 0.939692578522420840924574608221, 2.57984436195754145996936106048, 3.74406925621466086422400461978, 4.91739195384423799984246142862, 5.95304086450872112993551485224, 6.81122969450468860947435496042, 8.124228700984916440506368670452, 8.676705854091349691754905962220, 9.740877869908062319179838815976, 10.40463027497567232217690797027

Graph of the $Z$-function along the critical line